Analytical model of unsteady-state convective heat transfer between the heat carrier and the finite sizes plate adjusted for the thermal relaxation

Danila Makarushkin,Alexandr Kirsanov,Yury Kirsanov,Andrey Yudakhin,S.V Alekseenko

doi:10.1051/matecconf/201711502004

Danila Makarushkin, Alexandr Kirsanov + Show 3 more

Open Access

https://doi.org/10.1051/matecconf/201711502004

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Journal: MATEC Web of Conferences	Publication Date: Jan 1, 2017
License type: cc-by

Affiliation: Kazan Scientific Center

Abstract

A hyperbolic boundary value problem of the thermal conduction of a two-dimensional plate with the third kind boundary conditions is formulated. The transient thermal process in the plate is due to the temperature changes of the external medium over time and along the plate length, and also by a multiple step change of the plate surface heat transfer coefficient throughout the transient process. An analytical solution with improved convergence adjusted for thermal relaxation and thermal damping is obtained for the temperature field in the plate.

Highlights

IntroductionIn short-term transient processes, the results of calculations using the classical differential thermal conductivity equation wT wW a 'T (1)
In short-term transient processes, the results of calculations using the classical differential thermal conductivity equation wT wW a 'T (1)are characterized by large deviations from real processes due to neglect of thermal inertia
To overcome the limitations of Fourier formula q O grad T, Luikov [5], Tzou [6] et al proposed to take into account the effect of relaxation phenomena to heat conduction, due to which Fourier formula takes the form of expression, known as the Maxwell-Cattaneo-Luikov equation [5, 7, 8] or the dual-phase-lag Eq [6, 9, 10]: q Wq wq wW O grad T WT wT wW, (2)

Summary

Introduction

In short-term transient processes, the results of calculations using the classical differential thermal conductivity equation wT wW a 'T (1). Are characterized by large deviations from real processes due to neglect of thermal inertia This specificity of equation (1) was noted by such scientists as Maxwell [1], Onzager [2], Cattaneo [3], Vernotte [4], Luikov [5] et al, and was confirmed experimentally (Fig. 1). In this paper a plate of thickness 2h and length l is considered It is assumed, that the heat transfer coefficient of the plate varies in time stepwise, and the medium temperature (heat carrier) in a separate period depends on the time and length of the plate in the polynomial form: Tf , j Y ,t. ¦ ¦ Y ky k kt gk,lt l , k0 l0 where gk,l is the coefficient; t W Wq is the relative time; Y y l is the relative longitudinal coordinate; ky and kt are the counts of members of each row

The third kind hyperbolic boundary value problem

An analytic solution of the boundary value problem

Conclusions